SPHARMA approximations for stationary functional time series on the sphere

In this paper, we focus on isotropic and stationary sphere-cross-time random fields. We first introduce the class of spherical functional autoregressive-moving average processes (SPHARMA), which extend in a natural way the spherical functional autoregressions (SPHAR) recently studied in [8, 7]; more importantly, we then show that SPHAR and SPHARMA processes of sufficiently large order can be exploited to approximate every isotropic and stationary sphere-cross-time random field, thus generalizing to this infinite-dimensional framework some classical results on real-valued stationary processes. Further characterizations in terms of functional spectral representation theorems and Wold-like decompositions are also established.


Introduction
Over the last few years, the analysis of sphere-cross-time isotropic and stationary random fields has drawn a considerable amount of attention, due to strong motivations arising in Climate and Atmospheric Sciences, Geophysics, Astrophysics and Cosmology, and many other areas of research, see for instance [9,10,11,21,27,28,34] and the references therein. A lot of efforts has been spent, isotropic spherical random fields; under these circumstances it is possible to derive a double spectral decomposition based on spherical harmonics. Then, we focus on harmonic properties of functional autoregressive-moving average processes (see [3,8]), defined as random elements of L 2 (S 2 ) (as mentioned above, recently [8,7] have addressed the estimation of the functional linear autoregressive operators). Our approach is related to the extensive literature on linear stationary functional processes; however most of this literature is built on the estimation of autocovariance operators in the time domain, see among others [4,13,24].
After building this background, we arrive at our main results; as anticipated, we prove that, under very broad assumptions, any isotropic and stationary spherical process can be approximated arbitrary well by a SPHARMA model of sufficiently large degree. Our results can be viewed as the infinite-dimensional counterpart of the rational approximations of spectral densities and the Wold decomposition in the analysis of standard, real-valued stationary processes, see for instance [6,Chapters 4 and 5].
Plan of the paper. The paper is organized as follows. In Section 2, we describe the spectral properties of isotropic stationary sphere-cross-time random fields, while in Section 3 we provide a further characterization in terms functional spectral representation theorems. In Section 4, we introduce rigorously the SPHARMA class. Section 5 contains the main results of this work, that is, SPHARMA approximations of stationary and isotropic spherical random field, in terms of their harmonic transforms and in the L 2 (Ω) sense, together with a Wold-like decomposition. Lastly, Section 6 collects the proofs.
Notation. We will denote with B(S 2 ) the Borel σ-field on the unit sphere and with L 2 (S 2 ; C) := L 2 (S 2 , dx; C) the Hilbert space of square-integrable complex-valued functions on S 2 endowed with the usual inner product f, g L 2 (S 2 ;C) =´S 2 f (x)g(x)dx.
· L 2 (S 2 ;C) will be the norm induced by ·, · L 2 (S 2 ;C) ; to simplify the notation, sometimes we will replace the subscript L 2 (S 2 ; C) with 2. Moreover, the restriction of L 2 (S 2 ; C) to real-valued functions will be denoted by L 2 (S 2 ; R).
We will also use the same notation for L 2 (S 2 × S 2 ; C) and L 2 (S 2 × S 2 ; R). Let H be the Hilbert space of L 2 (S 2 ; C)-valued random elements with finite second moment, that is, f ∈ H is such that The associated inner product is defined as f, g H = E f, g L 2 (S 2 ;C) , for f, g ∈ H. For u, v ∈ L 2 (S 2 ; C), the tensor product u ⊗ v is defined to be the mapping that takes any element f ∈ L 2 (S 2 ; C) to u f, v ∈ L 2 (S 2 ; C). T TR is the trace (or nuclear) norm of the operator T , see [18]. For a real-or complex-valued function f defined on a set D, we define f ∞ := sup x∈D |f (x)|. δ b a is the Kronecker delta function.

Spectral characteristics
Consider the collection of random variables T (x, t), (x, t) ∈ S 2 × Z defined on the probability space (Ω, F, P). For every fixed t ∈ Z, T (x, t), x ∈ S 2 is a spherical random field as defined in [22,Chapter 5]; recall that we are implicitly assuming measurability with respect to the product For the rest of the paper, we are going to consider space-time spherical random fields which are jointly isotropic (in the spatial component) and stationary (in the temporal component). To this purpose, we give the following definition (see also [12]).
Thus, in this case we can define the autocovariance kernel at lag t which is also Hilbert-Schmidt, i.e. r t (·, ·) ∈ L 2 (S 2 × S 2 , R), and the corresponding operator R t : L 2 (S 2 ; C) → L 2 (S 2 ; C) induced by right integration, the autocovariance operator at lag t, In functional data analysis, it is usual to model random processes as random elements of some separable Hilbert space. Under the joint isotropy-stationarity assumption, the sequences of spherical random fields we are considering can be seen as a sequence of random elements of L 2 (S 2 ). More formally, for any fixed t ∈ Z, there exists a random element T t of L 2 (S 2 ) such that T (·, t) = for any x 0 ∈ S 2 . This implies that there exists a F-measurable set Ω ′ of P-probability 1 such that, for every ω ∈ Ω ′ , T (·, t, ω) is an element of L 2 (S 2 ).
Thus, {T t , t ∈ Z} is a stationary sequence of random elements in L 2 (S 2 ; R), with mean zero and E T 0 2 < ∞; see for instance [3,Definition 2.4] and R t coincides with the autocovariance operator defined as the Bochner integral In this setup, it is possible to show that the following representation holds in the L 2 (Ω) sense for every (x, t) ∈ S 2 × Z and in the L 2 (S 2 × Ω) sense for every t ∈ Z; the sequence {Y ℓ,m , ℓ ≥, m = −ℓ, . . . , ℓ} is a standard orthonormal basis for L 2 (S 2 ; C) of spherical harmonics, whereas, for fixed t ∈ Z, {a ℓ,m (t), ℓ ≥, m = −ℓ, . . . , ℓ} is a triangular array of zero-mean uncorrelated random coefficients defined as This result can be understood as a direct application of the spectral theorem for isotropic random fields on the sphere, see [22,Chapter 5 and in particular Theorem 5.13]. In this sense, it does not give insights on the temporal dynamics of the process and, hence, on its complete second-order structure.
Following [26], we shall use the conditions below to define the spectral density kernels and the spectral density operators and to prove part of our main results in Section 5. Under these conditions, we are also able to give first a Functional Cramér Representation which involves a L 2 (S 2 ; C)-valued orthogonal increment process, and then to obtain a double spectral representation with respect to both space and time, see Section 3 below. We stress that in this section and in Section 3, as in [25], it is not assumed any other prior structural properties for the stationary sequence (e.g., linearity or Gaussianity).

Condition 2.3.
For an isotropic stationary space-time spherical random field T (x, t), (x, t) ∈ S 2 × Z with continuous covariance kernels (2.1) on S 2 × S 2 , consider one of the following conditions: (i) the autocovariance kernels satisfy t∈Z r t 2 < ∞; (ii) the autocovariance operators satisfy t∈Z R t TR < ∞.
In [26] there is an extensive discussion on the role of such assumptions. Similarly here, under Condition 2.3 (i), it is possible to define the spectral density kernel at frequency λ ∈ [−π, π], where the convergence is in · 2 . It is uniformly bounded and also uniformly continuous in λ with respect to · 2 . The spectral density operator F λ : L 2 (S 2 ; C) → L 2 (S 2 ; C), the operator induced by the spectral density kernel through right-integration, is self-adjoint and nonnegative definite for all λ ∈ R. Moreover, the following inversion formula holds in the L 2 sense: Under Condition 2.3 (ii), we can define the spectral density operator at frequency λ ∈ [−π, π] where the convergence holds in nuclear norm. F λ is trace class and F λ TR ≤ t∈Z R t TR < ∞, λ → F λ TR is uniformly continuous and The reader is referred to [26] for proofs of these assertions. However, exploiting joint isotropy-stationarity of the space-time spherical random field, we can specialize all the previous results and obtain a neat expression for our quantities of interest. First of all, the sequence of zero-mean random coefficients satisfies (see Equation (6.2) in Proof of Proposition 3.1) and, as a consequence of Schoenberg's Theorem [31], the covariance kernel is shown to have a spectral decomposition in terms of Legendre polynomials, i.e., where ·, · denotes the standard inner product in R 3 , P ℓ (·) denotes the ℓ-th Legendre polynomial [32,Section 4.7] and the series is uniformly convergent.
Remark 2.6. Following the works [31] and [15], in [2] the authors give a mathematical characterization of covariance functions for isotropic stationary random fields over S 2 × R. In [12] the regularity properties of such covariance functions have been investigated for the case where a double Karhunen-Loève expansion holds. Examples of random fields satisfying this decomposition are found in the Appendix of [28].
As a consequence of (2. 3), that is, the Y ℓ,m 's are eigenfunctions of both R t and F λ , and the C ℓ (t)'s and f ℓ (λ)'s are the associated eigenvalues. Moreover, by the inversion formula (2.2), we have that f ℓ (λ) := 1 2π t∈Z e −itλ C ℓ (t). The eigenvalues f ℓ (λ) are also uniformly bounded and uniformly continuous in λ with respect to · 2 . Indeed, and, given ǫ > 0, there exists δ > 0 such that Clearly, under (ii), the trace class norm is given by

Spectral representations
This section builds on the earlier works [26,25] and it provides some results on a double spectral representation, with respect to both the temporal and spatial components of the field. The main purpose is to study these objects, trying to simultaneously capture the surface structure (spatial component) as well as the dynamics in time (temporal component); what in [25] is called within/between curve dynamics. We then specilize the results in [25], for dependent random functions on the interval [0, 1], to the case of the sphere, making this paper self-contained and complete.
The following proposition is the analogue of Theorem 2.1 in [25] and it can be seen as the infinitedimensional version of the well-known spectral representation of real-valued stationary processes.

The representation (3.1) is called the Cramér representation of T t , and the stochastic integral involved can be understood as a Riemann-Stieltjes limit, in the sense that
Now, we are going to establish a double spectral representation result, by showing the relation between the orthogonal increment process {α ℓ,m (λ), −π ≤ λ ≤ π} and {Z λ , −π ≤ λ ≤ π}. It is worth to notice that, under Condition 2.3, all the results presented in [25] can be easily extended to our framework, including the so-called Cramér-Karhunen-Loève Representation. Such a representation decomposes the space-time spherical random field into uncorrelated functional frequency components, exploiting an orthonormal basis for L 2 (S 2 ; C) made up of eigenfunctions of the spectral density operator F λ . However, in the anisotropic case, these eigenfunctions are unknown and have to be estimated. The stronger conditions allows to apply directly theorems from [25], since we have an explicit eigenvalue-eigenfunction decomposition of the spectral density operator in terms of spherical harmonics.

Remark 3.3. Note that the effective dimensionality of each frequency component is captured by the eigenvalues of the spectral density operators. The approximation error is then given by
see also [25,Remark 3.10].

Spherical functional ARMA
In this section, we extend the spherical functional autoregressions (SPHAR), first introduced in We first recall the definitions of spherical white noise and isotropic kernel operator, see [8,7].
{C ℓ;Z } denoting as usual the angular power spectrum of Z(·, t); spherical white noise if it satisfies (i) and the random fields Z(x, t), x ∈ S 2 , t ∈ Z, are independent and identically distributed.

Definition 4.2. A spherical isotropic kernel operator is an application
for some continuous k : The following representation holds in the L 2 -sense for the kernel associated with Φ: The coefficients {φ ℓ , ℓ ≥ 0} corresponds to the eigenvalues of the operator Φ and the associated eigenfunctions are the family of spherical harmonics {Y ℓ,m }, yielding Thus, it holds ℓ (2ℓ + 1)φ 2 ℓ < ∞, and hence this operator is Hilbert-Schmidt (see, e.g., [18]). In [8,7], the authors also consider trace class operators, namely, such that ℓ (2ℓ + 1)|φ ℓ | < ∞, for which the representation (4.1) holds pointwise for every x, y ∈ S 2 . Now, we focus on a space-time spherical random field T (x, t), (x, t) ∈ S 2 × Z , as defined in Section 1, for which it holds almost surely T (·, t) ∈ L 2 (S 2 ; R), t ∈ Z.

2)
for all (x, t) ∈ S 2 × Z, the equality holding both in the L 2 (Ω) and in the L 2 (S 2 × Ω) sense.  [8,7] introduce two estimation procedures for the spherical autoregressive kernels {k j , j = 1, . . . , p} and investigate asymptotic properties of the corresponding nonparametric estimators. Specifically, in [8], the authors focus on the solutions of a functional L 2 -minimization problem, while, in [7], they add a convex penalty term to study LASSO-type estimators under sparsity assumptions.
Now, define the polynomials φ ℓ : C → C and θ ℓ : note that the actual degrees can change with ℓ. Clearly, particular cases of the SPHARMA(p, q) process can be obtained by letting one of the two sequences constant and equal to 1. For instance, if φ ℓ (z) ≡ 1, for all ℓ ≥ 0, we obtain a spherical functional moving-average process of order q (or SPHMA(q)), whereas, If θ ℓ (z) ≡ 1, for all ℓ ≥ 0, then we have the so-called spherical functional autoregressive process of order p (or SPHAR(p)), see [8,7].
Under this assumptions, Condition 2.3 holds and the eigenvalues of the spectral density operators F λ are defined as in this case, Condition 4.6 simply becomes |φ ℓ | < 1, for all ℓ ≥ 0. Moreover, The proof of the following statement is given already in [3] for the simplest case of order one Hilbert-valued autoregressive processes, but here we construct explicitly the solution with a slightly different argument for completeness.

SPHAR and SPHMA approximations of spectral density operators
In what follows, we show that for any real-valued isotropic stationary random field, with spectral density kernels f λ satisfying Condition 2.3 (i), it is possible to find both a causal SPHAR(p) process and an invertible SPHMA(q) process whose spectral density kernels are arbitrarily close to f λ in the L 2 norm. This suggests that the original process can be approximated in some sense by either a SPHAR(p) or a SPHMA(q) process. Similar results hold for the spectral density operator F λ in the trace class norm under the stronger Condition 2.3 (ii). Below we will denote with T L a band-limited space-time spherical random field, namely such that it can be expanded in terms of finitely many spherical harmonics, up to a finite multipole ℓ = L.

The L 2 (Ω) approximations
Here we prove that any stationary spherical functional process can be approximated in the L 2 (Ω) sense by both a SPHMA process and a SPHAR process of sufficiently large degree. We also establish a functional Wold decomposition, which allows to represent the field as a sum of a linear process and a deterministic process, similarly to the finite-dimensional case, see [6,Theorem 5.7.1] and also [3,5]. This result is given in the auxiliary Lemma 5.6 and it is instrumental for the proof of our last main theorem.
More formally, consider as usual a zero-mean isotropic stationary process T (x, t), (x, t) ∈ S 2 × Z .
The following conditions will be used to state our second main result, with the additional Wold-like decomposition.
Condition 5.4. Consider the following assumptions:

For a purely non-deterministic process it is clear that the h-step prediction mean squared error
converges as h → ∞ to the total variance of the process. T (x, t), (x, t) ∈ S 2 × Z satisfying the hypotheses of Lemma 5.6 and Condition 2.3 (i) or (ii), the eigenvalues of the spectral operator F λ can be expressed as

Remark 5.8. For an isotropic stationary random field
We are now in the position to present our last main theorem. The statement makes precise the way in which the L 2 (Ω) approximations hold.
Theorem 5.9. An isotropic stationary random field T (x, t), (x, t) ∈ S 2 × Z satisfying Conditions 5.4 (i) and (ii) is such that, for all ǫ > 0, there exists integers L and q such that . Moreover, for all ǫ > 0, there exist integers L and p such that Both results also hold in the L 2 (S 2 × Ω) sense.

Proofs
Proof of Proposition 3.1. The proof follows the same lines of [25]. Let H be the Hilbert space of L 2 (S 2 ; C)-valued random elements with finite second moment and M 0 be the complex linear space spanned by all finite linear combinations of the T t 's, Let e t : ν → e itν , which belongs to the (complex) Hilbert space L 2 ([−π, π], F ν TR dν) endowed with the standard inner product F ν TR being the nuclear norm of the spectral density operator. Now, define the linear operator E by linear extension of the mapping T t → e t . E is well defined and a linear isometry; in particular, the inversion formula (2.2) gives Then, we extend its domain to M, the closure of M 0 in H (see [25] for further details); the extension has a well-defined inverse E −1 : L 2 ([−π, π], F ν TR dν) → M. For any ω ∈ (−π, π], we define Z ω = E −1 (½ [−π,ω) ) ∈ M and Z −π ≡ 0. By the isometry property, Hence, ω → Z ω is an orthogonal increment process. The proof follows with definition of an operator ζ as extension of the mapping The operator ζ is, by (6.1), an isomorphism with domain L 2 ([−π, π], F ν TR dν), and in addition ζ = E −1 . This in turn implies T t = E −1 (e t ) = ζ(e t ). If g is cadlag with a finite number of jumps, then ζ(g) is in fact the Riemann-Stieltjes integral (in the mean square sense) with respect to the orthogonal increment process Z ω : In conclusion, T t =´π −π e itλ dZ λ , as claimed.
We just need to prove that Recall that {α j , j ∈ Z} as defined in (6.3) represent the Fourier coefficients of the indicator function ½ [−π,λ) (·). Then, its k-th order Fourier series approximation is given by const uniformly over λ by assumption, it holds that By continuity of E, we conclude that in the L 2 -sense.
Moreover, see [25], Under isotropy this is equal to which means that The next proof is composed of two steps. First we show that the T (x, t) = lim k→∞ T k (x, t) is a solution of the SPHARMA(p, q) equation (4.2); and then we prove that any isotropic stationary solution of (4.2) takes the form (4.7).
Proof of Proposition 4.11. First note that, under Condition 4.6, for any ℓ ≥ 0, θ ℓ (z)/φ ℓ (z) has a power series expansion, that is, and, therefore, For ℓ ≥ 0, consider the stationary process here we take {ε(t), t ∈ Z} to be a white noise sequence with variance identically equal to one. The spectral density of {X ℓ (t), t ∈ Z} is given by (see [6]) Now, recall the identity Moreover, under Condition 4.6, for the non-degenerate polynomials it holds that which again is easily shown to be zero by lim k→∞ ∞ h=k−j+1 |ψ h;ℓ | 2 = 0 and Dominated Convergence Theorem. The argument involving the L 2 (S 2 × Ω) limit is analogous.
Proof of Theorem 5.2. This proof follows the same lines of the previous one. Here, we make use of [6,Theorem 4.4.3 and Corollary 4.4.2], which ensure that for ℓ = 0, . . . , L there exists an AR(p ℓ ) process with spectral densityf ℓ such that for all λ ∈ [−π, π].
We then take p = max ℓ≤L p ℓ .
Proof of Lemma 5.6. We just prove convergence in the L 2 (Ω) sense; the L 2 (S 2 × Ω) argument is analogous. First of all, by Theorem 5.7.1 in [6], we can deduce that, for fixed (ℓ, m), Proof of Theorem 5.9. Under Condition 5.4 (ii), the Wold decomposition has no deterministic component, that is, V (x, t) = 0 for all (x, t) ∈ S 2 × Z. Then, as a consequence of Lemma 5.6, for all ǫ > 0, there exists integers L and q such that Moreover, since a ℓ,m;Z (t) ∈ M ℓ,m;t for all ℓ, m, t, the representation can be inverted: with ∞ j=1 φ 2 ℓ;j < ∞. Then, for all ǫ > 0, there exist integers L and p such that Similarly, it is possible to show the L 2 (S 2 × Ω) approximation.